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"There are two ways to teach mathematics, namely the systematic way and the application-oriented way"- E. Zeidler

I'm a fresh researcher on PDEs, especially interested in evolution equations in abstract spaces, and recently switch to a more application-oriented field. My most frequently used tools are the ones from (linear or nonlinear) functional analysis and basic PDEs theory. When I engage deeply in studying some concrete PDEs problems, I need to study the quantitative properties of solutions or functions in some certain function spaces, which more or less related to harmonic analysis. Unfortunately, I do not have the chance to learn this tool systematically.

So my question is "Are there some monographs on harmonic analysis that put emphasis on applications to PDEs?", just like the "nonlinear functional analysis and its applications" by Zeidler to "functional analysis". Any suggestion on reading it/them will be appreciated. Although the theory of "pseudo-differential operators" stems from the distribution theory and Fourier analysis, I tend to exclude it from "harmonic analysis", since it is quite independent.

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    $\begingroup$ I think the books of Hörmander count as such... $\endgroup$
    – András Bátkai
    Commented Dec 2, 2016 at 8:55
  • $\begingroup$ Thank you. But I tend to believe that "the pseudo-differential operators theory" is an independent topic. Actually, I have learnt basic results of this theory. $\endgroup$
    – Ice sea
    Commented Dec 2, 2016 at 9:42

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This excellent book by Elias Stein gives you a rich resource on Harmonic Analysis and prepares for PDE needs. It's accessible online.

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H. Bahouri, J.-Y. Chemin, R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations is a good one.

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  • $\begingroup$ That is so great! Thank you. I have to say that is EXACTLY what I want. $\endgroup$
    – Ice sea
    Commented Dec 3, 2016 at 3:33

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